A 'factory' is a building or group of buildings ,where large amounts of goods are produced by using machines.
A 'plant' is also a factory ,but more specialized in either:
- making cars, equipments:
car assemby plant
or
- producing power or processing chemicals in particular:
a nuclear plant
a power plant
a chemical plant
-The British consider the 'plant' ( uncountable) also as the machines and equipments used in industry ( plant hire ).